STAT 1301

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STAT 1301

• Expected Value, Standard Error
• for
• AVERAGES

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RECALL

• Sampling without replacement from a population
  is often analogous to drawing with replacement
  from a box

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Recall We use sample statistics to estimate
population parameters

• We will study 2 types of population parameters in
  this course
• 1. averages
• 2. percents
• TODAY We focus on AVERAGES

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Chance Process

• Take a random sample from a population and find
  the Sample Average
• We use X (read X-bar) to denote the Sample
  Average

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Take a random sample of size n from a population

• Question
• What do we expect the Sample Average to be?
• Answer
• We expect it to be close to the population
  average (average of the box)

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Terminology

• The Population Average is called the EXPECTED
  VALUE of the Sample Average
• Notation EV(X) population average

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Figure 1
sample average
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• We DO NOT expect that X will be exactly equal
  to the population average
• More precisely, we state the situation as
  follows
• X Population Average Chance Error

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Figure 2
chance error
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How large do we expect the chance error to be?

• The likely size of the chance error is measured
  by the Standard Error of the Sample Average
• Notation SE(X)

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Standard Error of X

• SD of Population SE(X)
  sample size
• Note As sample size increases, SE(X)
  decreases .

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The fact that SE(X) decreases as sample size
increases is just common sense

• larger samples tend to result in more accurate
  estimates of the true population average
• (when we take a probability sample)

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Summary

• A sample average is likely to be around its
  Expected Value (i.e. the population average), but
  is likely to be off by a chance error similar in
  size to the Standard Error (i.e. SE( X).

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Another Point

• Approximately 95 of sample averages will be
  within 2 SE(X) of the population average
• Nearly all sample averages are within 3 SE(X) of
  the population average